https://file.aiquickdraw.com/mathbot/file/1734572927471xav1jr89.jpg

The question asks to verify if the set \(\{\mathbf{u}, \mathbf{v}\}\), where \(\mathbf{u} = (2, -3)\) and \(\mathbf{v} = (1, 2)\), forms a basis and then find the components of \(\mathbf{w} = (9, 4)\) in this basis.

### Solution:

#### a) Verifying if \(\{\mathbf{u}, \mathbf{v}\}\) is a basis:
To form a basis, \(\mathbf{u}\) and \(\mathbf{v}\) must be linearly independent. This happens if the determinant of the matrix formed by \(\mathbf{u}\) and \(\mathbf{v}\) is non-zero.

Matrix:
\[
\mathbf{A} =
\begin{bmatrix}
2 & 1 \\
-3 & 2
\end{bmatrix}
\]

Determinant:
\[
\text{det}(\mathbf{A}) = (2)(2) - (1)(-3) = 4 + 3 = 7 \neq 0
\]

Since \(\text{det}(\mathbf{A}) \neq 0\), the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are linearly independent and thus form a basis.

#### b) Finding the components of \(\mathbf{w}\) in the basis \(\{\mathbf{u}, \mathbf{v}\}\):
We express \(\mathbf{w}\) as a linear combination of \(\mathbf{u}\) and \(\mathbf{v}\):
\[
\mathbf{w} = c_1 \mathbf{u} + c_2 \mathbf{v}
\]
where \(c_1\) and \(c_2\) are scalars.

This translates to the system of equations:
\[
9 = 2c_1 + c_2
\]
\[
4 = -3c_1 + 2c_2
\]

Solving this system:
1. From the first equation:
\[
c_2 = 9 - 2c_1
\]

2. Substitute \(c_2\) into the second equation:
\[
4 = -3c_1 + 2(9 - 2c_1)
\]
\[
4 = -3c_1 + 18 - 4c_1
\]
\[
4 = 18 - 7c_1
\]
\[
7c_1 = 14 \implies c_1 = 2
\]

3. Substitute \(c_1 = 2\) into \(c_2 = 9 - 2c_1\):
\[
c_2 = 9 - 2(2) = 9 - 4 = 5
\]

Thus, the components of \(\mathbf{w}\) in the basis \(\{\mathbf{u}, \mathbf{v}\}\) are:
\[
\mathbf{w} = 2\mathbf{u} + 5\mathbf{v}
\]

### Final Answer:
- \(\{\mathbf{u}, \mathbf{v}\}\) forms a basis.
- \(\mathbf{w}\) in the basis \(\{\mathbf{u}, \mathbf{v}\}\) is \(2\mathbf{u} + 5\mathbf{v}\).

Would you like further details or explanations?

### Related Questions:
1. How can we determine if three or more vectors form a basis?
2. What is the geometric interpretation of linear independence and basis in 2D space?
3. How do we find the components of a vector in a non-standard basis in three-dimensional space?
4. Can this process be extended to matrices and higher-dimensional spaces?
5. Why is the determinant a reliable test for linear independence in 2D or 3D?

### Tip:
Always ensure the determinant of the basis matrix is non-zero to confirm linear independence before proceeding with computations.

Given the vectors \(\mathbf{u} = (2, -3)\), \(\mathbf{v} = (1, 2)\), and \(\mathbf{w} = (9, 4)\), verify that \(\{\mathbf{u}, \mathbf{v}\}\) forms a basis and write the components of \(\mathbf{w}\) in this basis.

Explore how to verify if \(\{(2, -3), (1, 2)\}\) forms a basis in \(\mathbb{R}^2\) and learn to decompose \((9, 4)\) in terms of this basis. This problem highlights key linear algebra concepts such as determinants, linear independence, and basis representation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation